If P `((a)/(3),4)` is the mid - point of the line segment joining the points Q(-6,5) and R(-2,3), then the value of a is

Options

A. -4
B. -12
C. 12
D. -6

Explanation

To find the value of \( a \) given that point \( P \left( \frac{a}{3}, 4 \right) \) is the midpoint of the line segment joining points \( Q(-6, 5) \) and \( R(-2, 3) \), we can use the midpoint formula.

### Step-by-step Solution:

1. **Identify the coordinates of points Q and R**:
- \( Q(-6, 5) \)
- \( R(-2, 3) \)

2. **Recall the midpoint formula**:
The midpoint \( M \) of a line segment joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

3. **Apply the midpoint formula to points Q and R**:
- For the x-coordinate:
\[ x_P = \frac{-6 + (-2)}{2} = \frac{-6 - 2}{2} = \frac{-8}{2} = -4 \]
- For the y-coordinate:
\[ y_P = \frac{5 + 3}{2} = \frac{8}{2} = 4 \]

4. **Set the coordinates of point P equal to the calculated midpoint**:
- We know \( P \left( \frac{a}{3}, 4 \right) \) is the midpoint, so we can set up the equations:
- For the y-coordinate:
\[ 4 = 4 \quad \text{(This is satisfied)} \]
- For the x-coordinate:
\[ \frac{a}{3} = -4 \]

5. **Solve for \( a \)**:
- Multiply both sides by 3:
\[ a = -4 \times 3 = -12 \]

6. **Conclusion**:
The value of \( a \) is \( -12 \).

### Final Answer:
\[ a = -12 \]